In recent years, there have been proposed and developed various automatic speed control technologies. An adaptive cruise control system, capable of executing a following control function, has been disclosed in Japanese Patent Provisional Publication No. 11-192858 (hereinafter is referred to as “JP11-192858”), corresponding to U.S. Pat. No. 6,311,120, issued on Oct. 30, 2001. In the automatic speed control device disclosed in JP11-192858, during a following control mode in which a host vehicle (an ACC vehicle) can automatically follow a preceding vehicle, while maintaining the host vehicle's distance from the preceding vehicle at a desired inter-vehicle distance, the device controls the host vehicle's speed so that the host vehicle is automatically accelerated up to a preset vehicle speed, when the preceding vehicle itself changes its traffic lane and disappears from the host vehicle's driving lane. On the other hand, when the preceding vehicle actually exists ahead of the host vehicle but an inter-vehicle distance sensor, employed in the host vehicle, loses the track of the preceding vehicle owing to various factors, for example, curves, slopes, weather or the like, the device disclosed in JP11-192858 maintains or holds the host vehicle's speed at a vehicle speed given at the time when the preceding vehicle has been lost, until such time that the host vehicle reaches a lost point where the preceding vehicle has been lost, so as to inhibit the automatic accelerating mode from being initiated and to prevent the host vehicle's distance from the preceding vehicle from being reduced inadvertently or undesirably.